Issuers of securities sell their debt securities to domestic and global capital markets. Debt security issuers generally create different types of debt securities that are with maturities across the yield curve. For example, debt security issuers may create both short-term debt securities with maturities of a year or less and long-term debt securities with maturities of over a year. Non-callable or “bullet” securities are attractive because of their liquidity, price transparency, and spread advantage relative to comparable U.S. Treasuries securities with similar maturity periods. Issuers of callable debt securities effectively buy a call option from investors and compensate those investors with additional yield above comparable bullet securities.
The three main structural characteristics of a callable debt security are the maturity date, the lockout period, and the type of call feature. The maturity date of a callable debt instrument is the latest possible date at which the security will be retired and the principal redeemed. The lockout period refers to the amount of time for which a callable security cannot be called. For example, with a 10 non-call 3-year (“10nc3”) debt security, the security cannot be called for the first three years. The call feature refers to the type of call option embedded in a callable security.
American-style callable debt is a debt security that has a continuous call feature after an initial lockout period. The investor is compensated for this type of call feature by receiving a higher yield in exchange for providing the issuer with the flexibility to call the security at any time after the lockout period with the requisite amount of notice. European-style call feature enables the issuer to exercise the option to call the debt on a single day at the end of the initial lockout period. European-style callable securities provide the investor an opportunity to obtain a greater spread over a comparable bullet security while reducing the uncertainty of a continuous call option. The Bermudan-style callable debt security is callable only on coupon payment dates, for example, semiannual dates after the conclusion of the initial lockout period.
With the rapid development of the fixed income securities market and the introduction of more complicated contingent claims, it is important to provide a general framework for describing interest rate movements and valuing interest rate products. By fitting a model to available interest rate data, one can discover the dynamics of term structure and the relationship between interest rates and derivative prices. By modeling interest rate securities, a financial institution is also able to manage the risk of its portfolio by determining the likely range of future prices and value at risk, among other variables.
Currently, the common numerical methods employed in derivatives valuation include the Monte-Carlo simulation method, finite difference algorithms and lattice approaches. Monte-Carlo simulation can be used for term structure modeling and derivatives pricing and can be applied to a variety of market instruments, such as various kinds of European-style callable debt securities. The main advantage of Monte-Carlo simulation is that it can, without much additional effort, incorporate complex payoff functions of complicated path-dependent securities. The computational costs for Monte-Carlo simulation increase linearly with the number of underlying factors. Therefore, Monte-Carlo simulation is more effective in valuing multi-factor models.
However, since the standard error of an estimate is inversely proportional to the square root of the number of simulation runs, a large number of simulation runs using the Monte-Carlo simulation method are generally required in order to achieve a desired level of precision. Since one cannot predict whether early exercise of an option call is optimal when a particular asset price is reached at a particular instant, it was a commonly held view that Monte-Carlo simulation could not be used to handle early exercise decisions for American-style derivative securities. Therefore, a number of advance approaches based on the Monte-Carlo simulation technique have been proposed for the valuation of American-style derivative securities.
Specifically, the Heath, Jarrow, and Morton approach represents a natural generalization of all existing non-arbitrage models. This approach is based on the specific non-arbitrage conditions imposed on the evolution of the forward rates and provides a unique martingale measure, under which, in general, the term structure evolution is not Markovian. However, the high computational costs involved with Monte-Carlo simulation is still an issue with all proposed approaches. Therefore, it is still difficult to use Monte-Carlo simulation for pricing American-style and Bermudan-style instruments, especially for a multi-factor specification.
Since a replicating portfolio can be found for every financial instrument, one can use non-arbitrage argument to derive a partial differential equation that describes the value of the financial instrument through time. One of the most efficient methods for solving partial differential equations is finite difference algorithms, which apply a discretization of the differential operators in the underlying equation. The numerical schemes arising from the discretization procedure can be broadly specified as either implicit or explicit schemes.
The lattice approach model is the simulation of the continuous asset price by a discrete random walk model. This model is the most widely used approach for valuing a wide variety of derivatives models because of its ease of implementation. In general, the lattice approach is equivalent to the explicit finite difference algorithms; however, it bypasses the derivation of partial differential equations and seeks to model the stochastic process directly, making it simpler to implement. Finite difference algorithms and lattice approach techniques may be easily used to price American-style securities. However, when the volatility of forward rates is not Markovian, the lattice approach leads to non-recombining trees, which is computationally restrictive for most practical applications.